v2009.01.01 - Convex Optimization
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v2009.01.01 - Convex Optimization

v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization

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110 CHAPTER 2. CONVEX GEOMETRY √ 2β svec S 2 + (a) (b) α [ α β β γ ] Figure 39: (a) Projection of the PSD cone S 2 + , truncated above γ =1, on αβ-plane in isometrically isomorphic R 3 . View is from above with respect to Figure 37. (b) Truncated above γ =2. From these plots we may infer, for example, the line { [ 0 1/ √ 2 γ ] T | γ ∈ R } intercepts the PSD cone at some large value of γ ; in fact, γ =∞ .

2.9. POSITIVE SEMIDEFINITE (PSD) CONE 111 2.9.2.1 rank ρ subset of the positive semidefinite cone For the same reason (closure), this applies more generally; for 0≤ρ≤M { A ∈ S M + | rankA=ρ } = { A ∈ S M + | rankA≤ρ } (198) For easy reference, we give such generally nonconvex sets a name: rank ρ subset of a positive semidefinite cone. For ρ < M this subset, nonconvex for M > 1, resides on the positive semidefinite cone boundary. 2.9.2.1.1 Exercise. Closure and rank ρ subset. Prove equality in (198). For example, ∂S M + = { A ∈ S M + | rankA=M− 1 } = { A ∈ S M + | rankA≤M− 1 } (199) In S 2 , each and every ray on the boundary of the positive semidefinite cone in isomorphic R 3 corresponds to a symmetric rank-1 matrix (Figure 37), but that does not hold in any higher dimension. 2.9.2.2 Subspace tangent to open rank-ρ subset When the positive semidefinite cone subset in (198) is left unclosed as in M(ρ) ∆ = { A ∈ S M + | rankA=ρ } (200) then we can specify a subspace tangent to the positive semidefinite cone at a particular member of manifold M(ρ). Specifically, the subspace R M tangent to manifold M(ρ) at B ∈ M(ρ) [160,5, prop.1.1] R M (B) ∆ = {XB + BX T | X ∈ R M×M } ⊆ S M (201) has dimension ( dim svec R M (B) = ρ M − ρ − 1 ) = ρ(M − ρ) + 2 ρ(ρ + 1) 2 (202) Tangent subspace R M contains no member of the positive semidefinite cone S M + whose rank exceeds ρ . Subspace R M (B) is a supporting hyperplane when B ∈ M(M −1). Another good example of tangent subspace is given inE.7.2.0.2 by (1876); R M (11 T ) = S M⊥ c , orthogonal complement to the geometric center subspace. (Figure 124, p.480)

110 CHAPTER 2. CONVEX GEOMETRY<br />

√<br />

2β<br />

svec S 2 +<br />

(a)<br />

(b)<br />

α<br />

[ α β<br />

β γ<br />

]<br />

Figure 39: (a) Projection of the PSD cone S 2 + , truncated above γ =1, on<br />

αβ-plane in isometrically isomorphic R 3 . View is from above with respect to<br />

Figure 37. (b) Truncated above γ =2. From these plots we may infer, for<br />

example, the line { [ 0 1/ √ 2 γ ] T | γ ∈ R } intercepts the PSD cone at some<br />

large value of γ ; in fact, γ =∞ .

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